1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Infinite series - Inverse trigonometry

  1. Aug 18, 2013 #1
    1. The problem statement, all variables and given/known data
    The sum of the infinite terms of the series
    is equal to

    Ans: B

    2. Relevant equations

    3. The attempt at a solution
    The given series can be written as
    $$\sum_{r=1}^{\infty} \text{arccot}\left(r^2+\frac{3}{4}\right)$$
    Simplifying the term inside the summation, I rewrite it as
    I don't see how to proceed further. I am thinking of converting the above in the form ##\arctan\left(\frac{x-y}{1+xy}\right)##, rewrite it in the form ##\arctan(x)-\arctan(y)## and perform the summation but I can't see a way to do this.

    Any help is appreciated. Thanks!
    Last edited: Aug 18, 2013
  2. jcsd
  3. Aug 18, 2013 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    I haven't worked it out, but I think you're on the right track. Try rewriting it as
    $$r^2 + \frac{3}{4} = \left(r^2 - \frac{1}{4}\right) + 1 = \left(r+\frac{1}{2}\right)\left(r-\frac{1}{2}\right) + 1$$
  4. Aug 18, 2013 #3
    Wow. That's great. Thanks a lot vela! :smile:

    I did try rewriting 3/4 as 1-1/4 but then I didn't notice that it could simplified to (r-1/2)(r+1/2). :redface:
  5. Aug 19, 2013 #4


    User Avatar
    Homework Helper

    This is one of the most sadistic questions I've seen. Where are you getting these questions?
  6. Aug 19, 2013 #5
    From practice sheets. Do you want me to send the links to those practice sheets by PM? I don't know if its ok to post them here.
  7. Aug 19, 2013 #6


    User Avatar
    Homework Helper

    A PM will be fine, thanks. I haven't seen these type questions before, is all.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted